cp's OEIS Frontend

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

Also number of cycle types of odd permutations.

Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006

Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:

(11) (111) (211) (221) (222) (331) (2222)

(1111) (2111) (2211) (2221) (3221)

(11111) (3111) (3211) (3311)

(21111) (22111) (22211)

(111111) (31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:

(2) (3) (4) (5) (6) (7) (8)

(31) (32) (33) (43) (44)

(41) (42) (52) (53)

(51) (61) (62)

(411) (322) (71)

(421) (422)

(511) (431)

(521)

(611)

(5111)

(End)

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.

a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006

Number of partitions of n where all parts (except for possibly the first part) are even; see example. - Joerg Arndt, Apr 22 2013

For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p. For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Oct 28 2021: (Start)

If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.

Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:

11 111 22 221 33 331 44 441 55

1111 11111 2211 22111 2222 22221 3322

111111 1111111 3311 33111 4411

221111 2211111 222211

11111111 111111111 331111

22111111

1111111111

These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.

(End)

Also number of partitions of n into exactly 4 parts.

Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018

From Gus Wiseman, Jun 27 2021: (Start)

Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:

(4) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6)

(5,2,1) (5,3,2) (5,4,3) (6,5,3) (7,6,3)

(6,3,1) (6,4,2) (7,5,2) (8,6,2)

(7,4,1) (8,5,1) (9,6,1)

(6,3,2,1) (6,4,3,1) (6,5,4,1)

(7,4,2,1) (7,4,3,2)

(7,5,3,1)

(8,5,2,1)

(6,4,3,2,1)

(End)

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even terms of a finite sequence of positive integers.

For the specified value of n, the second Maple program lists the partitions of n counted by a(n).

Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - Gus Wiseman, Jul 26 2021

It follows by conjugation that the partition statistics "alternating sum" and "number of odd parts" are equidistributed. Consequently, the self-conjugate partitions satisfy the required condition.

In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, OP gives the number of odd terms of a finite sequence of positive integers.

For the specified value of n, the second Maple program lists the partitions of n counted by a(n).

Number of integer partitions of n with the same number of odd parts as their conjugate. - Gus Wiseman, Jun 27 2021

Also the number of strict integer partitions of 2n with reverse-alternating sum >= 0.

Also the number of reversed strict integer partitions of 2n with alternating sum >= 0.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.